¨¸Ó³ ¢ —Ÿ. 2002. º 6[115]
Particles and Nuclei, Letters. 2002. No. 6[115]
“„Š 539.172.17
NEUTRON DRIP LINE
IN THE REGION OF OÄMg ISOTOPES
Yu. S. Lutostansky a , S. M. Lukyanov b , Yu. E. Penionzhkevich b , M. V. Zverev c
a
Moscow Physical Engineering Institute, Moscow
b
Joint Institute for Nuclear Research, Dubna
c
Russian Research Centre ®Kurchatov Institute¯, Moscow
The results of the last experiments (RIKEN, GANIL, Dubna) on the mapping of the neutron drip
line in the oxygenÄmagnesium region are discussed. The analysis of stability and properties of the
neutron drip-line isotopes from the OÄMg region are performed using self-consistent theory of ˇnite
Fermi systems (STFFS). Conclusion about the vanishing of N = 20 and N = 28 and appearance of the
new nuclear magic numbers N = 16 and N = 26 is given.
¡¸Ê¦¤ ÕÉ¸Ö ·¥§Ê²ÓÉ ÉÒ ¶µ¸²¥¤´¨Ì Ô±¸¶¥·¨³¥´Éµ¢ (RIKEN, GANIL, „Ê¡´ ) ¶µ ¨§ÊÎ¥´¨Õ
´¥°É·µ´´µ-¨§¡ÒɵδÒÌ Ö¤¥· ¢¡²¨§¨ £· ´¨Í ¸É ¡¨²Ó´µ¸É¨ ¤²Ö ¨§µÉµ¶µ¢ ±¨¸²µ·µ¤ , Ëɵ· , ´¥µ´ ,
´ É·¨Ö ¨ ³ £´¨Ö. ´ ²¨§ ¸¢µ°¸É¢ ÔÉ¨Ì Ö¤¥· ¢Ò¶µ²´¥´ ¢ · ³± Ì ¸ ³µ¸µ£² ¸µ¢ ´´µ° É¥µ·¨¨ ±µ´¥Î´ÒÌ Ë¥·³¨-¸¨¸É¥³ (STFFS). ‘¤¥² ´µ § ±²ÕÎ¥´¨¥ µ¡ ¨¸Î¥§´µ¢¥´¨¨ ´¥°É·µ´´ÒÌ µ¡µ²µÎ¥± N = 20
¨ N = 28 ¢ Ôɵ° µ¡² ¸É¨ Ö¤¥· ¨ ¶µÖ¢²¥´¨¨ ´µ¢ÒÌ N = 16 ¨ N = 26.
INTRODUCTION
The study of properties of neutron-rich nuclei far from stability is one of the most
interesting areas of modern research in nuclear physics. The progress in our knowledge of
the properties of these nuclei has enormously broadened because of the new radioactive ion
beam facilities and the development of very sophisticated fragment separator.
An interesting feature has been found in the light neutron-rich nuclei. Neutron-rich
unstable nuclei with N ≈ 20 constitute a good example since establishment in 1975 [1].
The nuclei in this region are spectacular examples of shape coexistence between spherical
and deformed conˇgurations. For example, 31 Na and 32 Mg with N = 20 were found to be
deformed. This new region of deformation for N = 20−28 neutron-rich nuclei was discussed
in 1987 [2], while analyzing the sodium isotopes properties. It was shown that deformation
mixed single-particle neutron levels, destroyed standard shell structure and changed strongly
neutron separation and binding energy of nuclei. As a result, the drip line in the region
N = 20−28 is shifted by several mass units compared to expected spherical case.
Recent experiments at GANIL, MSU and RIKEN have been devoted to studying the
stability of the neutron-rich nuclei with Z > 7 and around N = 20. The variation of the shell
gap and deformation as a function of N and Z could be a major challenger.
Among recent results of the neutron drip-line study in the region of elements from O
to Mg, one can mention the experiments on the particle instability of neutron-rich oxygen
isotopes 26,27,28 O [3Ä6] and the discovery of particle stability of 31 Ne [7] and 31 F [8].
Neutron Drip Line in the Region of OÄMg Isotopes 87
The appearance of the so-called ®island of inversion¯ with respect to the particle stability
of isotopes has been claimed through various theoretical predictions. In neutron-rich sodium
isotopes this inversion was predicted and analyzed in [2]. A particular feature of nuclei in
this region is the progressive development of the static deformation in spite of the expected
effect of spherical stability due to the magic neutron number N = 20 [11, 12]. However, the
situation with N = 20 drip-line nuclei is not so clear because there may be other physical
reasons for destroying the magic besides the deformation.
A more clear picture is in N > 20 area. It was argued that the deformation might lead
to enhanced binding energies in some of yet undiscovered neutron-rich nuclei. The particle
stability of 31 F gives a strong evidence of the onset of the deformation in this region. One
may expect that the drip line for the uorineÄmagnesium elements could move far beyond the
presently known boundaries. Therefore, there is a great interest in studying of nuclei in the
region of neutron closure N = 28. Experimentally, the properties of 44 S had been studied
[13] and it was concluded there that the ground state of 44 S is deformed. This result suggests
a signiˇcant breaking of the N = 28 closure for nuclei near 44 S.
In this article we analyze also the nuclei near the neutron drip line in theoretical approach
and compare our previous calculations [16] proposed in spherical self-consistent basis with
analogous calculations made including deformation effects.
1. EXPERIMENTAL MAPPING OF THE NEUTRON DRIP LINE
New attempts to determine the neutron drip line for the FÄNeÄMg isotopes in the region of
the neutron numbers N = 20−28 were presented recently in the framework of RIKENÄDubna
and GANILÄDubna collaborations [14, 15] using the fragment spectrometers: LISE 2000 at
GANIL (France) and RIPS at RIKEN (Japan).
In particular, experiments were devoted to the direct observation of the 31 F, 34 Ne, 37 Na
and 40 Mg nuclei among the products of a primary beam fragmentation. In both experiments
the neutron-rich nuclei were searched for among the fragmentation products of 48 Ca beams
on tantalum targets. This neutron-rich beam and target were chosen to optimize the production
rate of the drip-line nuclei. The world records of mean beam intensity of the 48 Ca beam at
values above 150 pnA were reached at these experiments.
The reaction fragments were collected and analyzed by the fragment spectrometers operated in an achromatic mode and at the maximum values of momentum acceptance (5 %) and
maximum solid angles. To reduce the overall counting rate due to light isotopes, a wedge
was placed at the momentum dispersive focal plane.
The standard identiˇcation method of the fragments via time of ight (ToF), energy loss
(dE) and total kinetic energy (TKE) was used. A multiwire proportional detector was placed
in the dispersive plane spectrometer. This detector allowed one to measure the magnetic
rigidity of each fragment via its position in the focal plane, improving the mass-to-charge
resolution (A/Q).
The results of these experiments are shown in Fig. 1. Isotopes 34 Ne and 37 Na were
unambiguously identiˇed by measuring the magnetic rigidity, time of ight, energy loss and
total kinetic energy. So, both experiments [14, 15] give the very clear evidence for the particle
stability of 34 Ne, 37 Na.
88 Lutostansky Yu. S. et al.
Fig. 1. Two-dimensional A/Z versus Z plot, which was obtained in the reaction of the 48 Ca beam:
a) at GANIL using the LISE-2000 spectrometer; b) at RIKEN using the RIPS spectrometer. Isotopes
34
Ne and 37 Na were unambiguously identiˇed by measuring the magnetic rigidity, time of ight, energy
loss and total kinetic energy. So, both experiments [14, 15] give the very clear evidences of the particle
stability of 34 Ne, 37 Na
It was also found that particle instability of 33 Ne, 36 Na and 39 Mg was veriˇed by the low
background level.
The most interesting nuclide in this region is 40 Mg, which is probably not bound, since
no counts have been observed. The upper limit for the production cross-section of 40 Mg to
be less than 0.01 pb.
2. THEORETICAL APPROACH
From the theoretical point of view, the description of the light nuclei in the sd−pf shells
is still a problem. Various theoretical calculations (viz. ˇnite-range liquid drop (FRLD)
model [16], two versions of the shell model (SM) [17, 18], relativistic mean ˇeld theory [19], HartreeÄFock (HF) model [20] and relativistic Hartree + Bogoliubov (HF + B)
description [21]) exist and predict the position of the neutron drip line in this region. For
instance, the FRLD model gives a very strong binding energy for 40 Mg. In the framework
of this model, one- and two-neutron separation energies are even above 3.4 MeV. One may
notice that the FRLD model gives correct predictions for stability of 31 Ne and 31 F, implying
nuclear deformation effects for both the macroscopic and microscopic parts. According to
the shell model predictions [17], the last bound isotopes are 24 O, 27 F, 34 Ne, 37 Na, and 38 Mg.
However, slight changes of the drip line cannot be excluded since 37 Na was predicted to
be bound only by 250 keV, while 31 F and 40 Mg are unbound by 145, 470 and 550 keV,
respectively. According to another shell model calculation [18], 26 O, 34 Ne and 40 Mg are
the last stable isotopes against two-neutron emission, as indicated by their maximal binding
Neutron Drip Line in the Region of OÄMg Isotopes 89
Energies of separation of one (Sn ) and two (S2n ) neutrons for nuclei near the neurtron-stability
boundary calculated in STFFS spherical basis
Z
Nucleus
8
24
9
29
10
32
11
37
12
42
O
F
Ne
Na
Mg
N
16
20
22
26
30
Sn , MeV
3.59
1.64
1.65
0.99
1.13
S2n , MeV
Nucleus
5.85
26
0.90
31
O
N
Sn , MeV (S2n < 0)
18
0.69
F
22
<0
0.33
34
Ne
24
0.16
0.07
39
Na
28
<0
0.38
44
Mg
32
0.91
energy. Both SM and HF calculations for even-mass O, Ne and Mg indicate a disappearance
of shell magic numbers, and suggest an onset of the deformation and shape coexistence in
this region.
We calculated properties of neutron drip-line isotopes from OÄMg region using selfconsistent theory of ˇnite Fermi systems (STFFS), speciˇed by the quasiparticle Lagrangian
which is a functional of four normal and one anomalous densities [22Ä25]. This Lagrangian is
constructed in such a way that energy- and velocity-dependent two- and three-particle effective
forces are taken into account in effective-range approximation. We use the approximation
of volume pairing since all the characteristics under study, including the neutron-separation
energy, are determined mainly by the gap operator ∆ diagonal elements which are weakly
sensitive to the type of pairing [23, 24].
The results of STFFS calculations in spherical basis are shown in the table, which presents
neutron separation energies Sn and S2n calculated for extremely heavy stable isotopes that
are N -even and the one-neutron-separation energies Sn found for the next N -even isotope.
We can see that for evenÄeven nuclei the position of the boundary in the considered region is
determined by two-neutron instability. And as was predicted in [22] for 26 O, a quasistationary
state in 24 O + 2n system may exist as 26 O∗ with the width Γ ∼ 10−3 MeV. Analogously, for
other extra drip-line nuclei of A (evenÄeven) type, A∗ = A + 2n quasistationary systems may
exist, for example, 36 Ne∗ and 40 Mg∗ . The lifetime of such systems τ ≥ 10−18 s is much
more than inner nuclear time τnucl ∼ 10−22 s and neutron pair should be emitting in a weakly
bound di-neutron state.
From the table we can see that the results only for 24 O and 37 Na isotopes correspond to
the experimental data. That is because of the fact that 24 O isotope is spherical one and for
37
Na due to the deformation effects the neutron drip-line number until β2 < 0.4 is N = 26,
which corresponds particularly to the spherical basis calculations (see Fig. 3 and following
discussion).
Deformation in STFFS may be taken into account as in [25] for axial type deformation
for heavy and middle-mass nuclei, and in general as in [26], using very complicated energydensity functional approach. Here we estimate the inuence of the quadrupole deformation
using the approximation ˇrstly mentioned in [2] (the details of the developed method will
be soon published). In this method we estimate corrections δS2n and δSn as a function
of quadrupole deformation parameter β20 = β2 . At the ˇrst stage, we calculate energetic
parameters, single-particle levels ελ and wave functions in STFFS. At the second stage,
90 Lutostansky Yu. S. et al.
we calculate the inuence of deformation on energetic parameters increasing β2 value step
by step using ∆β2 . In this approach we must calculate the single-particle levels scheme
and wave functions, using an axially symmetric SaxonÄWoods potential with the
ˇxed parameter β2 = β i + ∆β2 . Getting new wave functions and ελ , we may
construct all necessary densities on axially
symmetric basis. Solving STFFS equations
with different values of parameter β2 , we
ˇnd energetic parameters of nuclide and
binding energy as functions of quadrupole
deformation. Such calculations were performed earlier for sodium isotopes including 35 Na, and the minimum of the binding
energy, corresponding to the ˇxed β2 parameter, was found.
To analyze the role of deformation in
our
region of drip-line nuclei, we use here a
Fig. 2. Calculated single-particle levels scheme and
simpler
estimation. We calculate our nuclei
wave functions, using an axially symmetric SaxonÄ
in
STFFS
spherical basis as zero approxiWoods potential for different values of parameter β2
mation and then we calculate their singlefor 37 Na
particle levels scheme and wave functions,
using an axially symmetric SaxonÄWoods potential for different values of parameter β2 .
One of the results of such calculations of ελ = ελ (β2 ) for 37 Na presented in Fig. 2 will be
discussed later.
3. DISCUSSION
For the neutron-rich oxygen isotopes, deformation effects are negligible and we calculate
them as spherical. Nevertheless, we analyzed deformation effects in Z = 8 region. According
to our estimations, when deformation is small (β2 < 0.4), good ®magic¯ number for oxygen
neutron-rich isotopes is N = 16. But in case of strong deformation with β2 > 0.4 the inverse
level 1/2− , splitting down from f7/2 state, may form isomers with N = 18. So magic
number N = 20 is destroyed by the deformation. On the other hand, according to spherical
STFFS calculations [22], strong pairing is developed in neutron system with N near 20 for
very neutron-rich nuclei. This could be the main reason for destroying the magic N = 20 in
the considered region of nuclei.
Interesting picture can be seen for Z > 8 drip-line nuclei. For uorine and neon dripline isotopes with deformation 0.2 < β2 < 0.4, the drip-line neutron number is N = 24
corresponding to 33 F and 34 Ne. This means that an addition of only one Z-unit to Z = 8
shifts the N drip line by eight neutrons, as illustrated on the map of nuclides in Fig. 3.
Comparing with the results of spherical calculations from the table, we can see in Fig. 3 that
due to deformation four neutrons are added to uorine and two neutrons to neon drip-line
isotopes. The question of 36 Ne particle stability is still open in our approach because of
uncertainties in the deformation and energetic parameters we use for β2 > 0.4.
Neutron Drip Line in the Region of OÄMg Isotopes 91
For the deformed sodium and magnesium, the neutron drip-line number is N = 26, which
corresponds to 37 Na and 38 Mg isotopes. As we can see in Fig. 2, N = 26 looks like a
new neutron magic number in some intervals of quadrupole deformation. Figure 2 illustrated
wide splitting of the single-particle states with increasing deformation and ελ (β2 ) dependence
differs greatly for different λ. In the case of isotopes with N = 26, 28 the 1f7/2 state
splitting is the most interesting. We can see that upper 7/2− level occupied by N = 27, 28 is
growing rapidly together with β2 and after β2 > 0.15 drip-line nuclides with N > 26 become
unbound, for example, 40 Mg and 39 Na (until β2 < 0.5). Thus, we can explain the strong
discrepancy with the table for heavy magnesium isotopes calculated there in spherical basis.
Fig. 3. The part of nuclide chart at the region neutron number N = 20 and N = 28. The deformed
nuclei are italicized and spherical nuclei are in roman type. The neutron drip line is shown by solid line.
Nuclei (∗ ) are predicted as quasistationary states (A + 2n)-type with a lifetime τ ≥ 10−18 s, emitting
neutron pair in a weakly bound di-neutron state
On the other hand, the dependence of 5/2− level occupied by N = 26 on deformation
parameter β2 is very weak in this case. But after β2 > 0.4 the 3/2+ level is rising up rapidly
from 1d3/2 state and N = 24 becomes the drip-line neutron number in small interval of
deformation parameters. For example, 33 F may exist as a very deformed nuclide. The case
of strong deformations with the parameter β2 > 0.5 is not discussed in this work because
self-consistent theoretical approach becomes very complicated here, where various levels
(1/2−, 3/2+, 5/2− ) are mixing in a very small energetic interval ∆ε ∼ 1 MeV for weakly
bound system. So the question of particle stability for very deformed 33 F and 36 Ne is still
open.
We analyzed the problem of N = 28 magic number from two points of view: from the
position of deformation destroying, as was described, the magic N = 28 in our region of
nuclei, and from the position of developed pairing in the neutron-rich nuclei. According to
STFFS spherical calculations [22] for N = 28 nuclei from 48 Ca to 47 K, the spacing energy
∆ε between the one-particle neutron state f7/2 , which is the highest occupied state in the
scheme without pairing, and the unoccupied state p3/2 is rather large (∆ε ∼ 2.5 MeV). The
gap operator matrix elements is small for these states (∆λ ∼ 0.3 MeV); therefore, there
is no developed pairing in these nuclei. However, for smaller Z numbers, the situation
has strongly changed, and for 46 Ar the ∆λ value is more than twice larger, and the level
spacing ∆ε decreases as we approach the neutron-stability boundary and falls below 1 MeV
92 Lutostansky Yu. S. et al.
at the drip line. The corresponding matrix elements grow up to ∆λ ∼ 1.2 MeV. Therefore,
the number N = 28 loses its magic properties in MgÄAr region up to the neutron drip
line.
SUMMARY
Nowadays the neutron-rich isotopes 34 Ne and 37 Na have been experimentally observed
in the reactions 48 Ca + nat Ta [14, 15]. Thus, most probably, the neutron drip line has been
reached for the neon and sodium isotopes. However, it seems that to deˇnitely conclude
whether these isotopes mark the drip line, further experimental efforts are needed. The
neutron-rich 40 Mg with N = 28 has not been observed in these experiments, which can be
explained by the deformation effects near the neutron drip line, destroying neutron shell with
N = 28.
According to our microscopic self-consistent calculations with the deformation and pairing
effects included, the new neutron drip line was established, and we can see that N = 20 and
N = 28 lose their magic properties for very neutron-rich isotopes up to the N drip line in
the OÄMg region. Neutron drip-line number for oxygen neutron-rich isotopes is N = 16.
For a certain region of the quadrupole deformation parameter β2 values, the neutron number
N = 26 is a new neutron drip-line number in the considered region of the nuclear map.
On the basis of previous calculations, it was predicted that quasistationary state in 24 O+2n
system may exist as 26 O∗ with the width Γ ∼ 10−3 MeV. Analogously, 36 Ne∗ = 36 Ne + 2n
and 40 Mg∗ = 38 Mg + 2n extra drip-line nuclear quasistationary system may exist with a
lifetime τ ≥ 10−18 s, emitting neutron pair in a weakly bound di-neutron state.
Acknowledgements. The authors would like to express their gratitude to the members
of the FLNR (JINR, Dubna)ÄGANIL (France) and FLNR (JINR, Dubna)ÄRIKEN (Japan)
collaborations for the fruitful experimental efforts and discussions of the experimental results
obtained in the joint experiments. This work has been carried out with ˇnancial support
from PICS-(IN2P3) No. 1171, grant from INTAS 00-00463 and grant No. 96-02-17381a of
the Russian Foundation for Basic Research.
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Received on November 5, 2002.